On the use of wavelets as an alternative to Fourier analysis for transient dynamic data

Various dynamic data analysis and synthesis tools for one-dimensional time- domain signals are employed to determine the frequency content of a signal for mechanical analysis. When tied to the fundamental frequency of the various com- ponents comprising the machinery being evaluated, this information gives an indication of the state or health of the machine. Current techniques for evaluating dynamic data for potential mechanical problems are primarily centered around the Fast Fourier Transform (FFT) and the Short-Time Fourier Transform (STFT).

The use of Fourier analysis for frequency component extraction is restricted to bandlimited stationary signals. Because of this stationary requirement, small transients may not be detected due to a smoothing effect of the FFT, or the FFT spectrum may be smeared due to frequency ramping and abrupt incidents or dis- continuities in the signal. Various techniques have been employed to overcome the limitations of the FFT for non-stationary data. These techniques range from win- dowed FFT (Gabor or Short-Time Fourier Transform), synchronous sampling to remove RPM-ramp effects, Wigner-Ville analysis, and, more recently, wavelet analysis. The primary goals of this thesis are to evaluate and compare various dynamic data analysis and synthesis tools, and to apply them to a few selected signal types.

The research consisted of several phases. First, Fourier analysis is described as it applies to one-dimensional time domain signals. The problems and inadequacies of the FFT analysis are illustrated. With this as a guide, the wavelet alternative to Fourier analysis is described in general, but with particular emphasis on the frequency interpretation of the wavelet analysis. A direct comparison of the wavelet and FFT for stationary data was performed to show that both approaches produce similar results for that regime. The analysis continues with a shift from station- ary data to varying degrees of non-stationary data to evaluate the degradation of the FFT and find limits to its applicability. Finally, the use of wavelets for non- stationary data is illustrated with mechanical engineering examples and compared to the STFT.

The theory, interpretation, and short comings of the FFT analysis were de- scribed and limits to the application of the FFT for non-stationary signals were established for the two major non-stationary characteristics of interest to the au- thor, frequency ramps and sudden impulses. It was shown that the FFT, with appropriate bin spacing and oversampling, provides adequate detection and useful information for signals which ramp in frequency, and does so much better than the wavelet approach, even though the time element of the signal is lost in the trans- formation. In addition, it was shown that the STFT provided superior results for the analysis of a complex signal as compared to the wavelet analysis for the basis functions used. However, it was also shown that the FFT is inadequate to describe a signal containing spikes since the FFT noise floor for a spike cannot be differentiated from the noise floor of a typical real spectrum. The wavelet has the capability to provide the detection of discontinuities in a signal, and therefore is superior to the FFT for such signals, and was shown to be a significant improvement for pulse-echo measurements due to this ability.